Download Inference for Quantitative Variable: Hypothesis Tests

Lecture 25: more Chapter 10, Section 1
Inference for Quantitative Variable:
Hypothesis Tests
z
Test about Population Mean: 4 Steps
Examples: 1-sided or 2-sided Alternative
Relating Test and Confidence Interval
Factors in Rejecting Null Hypothesis
Inference Based on t vs. z
©2011 Brooks/Cole, Cengage
Learning
Elementary Statistics: Looking at the Big Picture
1
Looking Back: Review

4 Stages of Statistics




Data Production (discussed in Lectures 1-4)
Displaying and Summarizing (Lectures 5-12)
Probability (discussed in Lectures 13-20)
Statistical Inference





©2011 Brooks/Cole,
Cengage Learning
1 categorical (discussed in Lectures 21-23)
1 quantitative: confidence intervals, hypothesis tests
categorical and quantitative
2 categorical
2 quantitative
Elementary Statistics: Looking at the Big Picture
L25.2
Behavior of Sample Mean (Review)
For random sample of size n from population
with mean and standard deviation ,
sample mean has
 mean
 standard deviation
 shape approximately normal for large
enough n
If is known, standardized follows
z (standard normal) distribution
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.4
Hypothesis Test About
(with z)
Problem Statement
1.
2.
3.
4.
Consider sampling and study design.
Summarize with , standardize to
assuming
is true; is z “large”?
Find P-value (prob. of Z this far
above/below/away from 0); is it “small”?
Based on size of P-value, choose
or
.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.5
Hypothesis Test About
1.
2.
3.
4.



with z (Details)
Consider sampling and study design.
Summarize with , standardize to
assuming
is true; is z “large”?
Find prob. of z this far above/below/away from 0
(P-value); consider if it is “small”.
Based on size of P-value, choose
or .
If sample is biased, mean of is not .
If pop<10n, s.d. of is not
.
If n is too small, distribution of
is not normal,
won’t standardize to z: graph data, see guidelines.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.6
Hypothesis Test About
1.
2.
3.
4.

with z (Details)
Consider sampling and study design.
Summarize with , standardize to
assuming
is true; is z “large”?
Find prob. of z this far above/below/away from 0
(P-value); consider if it is “small”.
Based on size of P-value, choose
or .
Assess P-value based on form of alternative
hypothesis (greater, less, or not equal)
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.7
Hypothesis Test About
with z (Details)
Alternative “>”: P-value is right-tailed probability
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.8
Hypothesis Test About
with z (Details)
Alternative “<”: P-value is left-tailed probability
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.9
Hypothesis Test About
with z (Details)
Alternative “≠”: P-value is two-tailed probability
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.10
Example: Assumptions for z Test



Background: Earnings of 446 surveyed university
students had mean $3,776. The mean of earnings for
the population of students is unknown. Assume we
know population standard deviation is $6,500.
Question: What aspect of the situation is unrealistic?
Response:
Looking Ahead: In real-life problems, we rarely know the value
of the population standard deviation. Eventually, we’ll learn how
to proceed when all we know is the sample standard deviation s.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.12
Example: Test with One-Sided Alternative



1.
2.
3.
4.
Background: Earnings of 446 surveyed university students
had mean $3,776. Assume population s.d. $6,500.
Question: Are we convinced that
is less than $5,000?
Response: State
Data production issues were discussed for confidence interval.
Output shows sample mean ______ and z = ____. Large?_____
P-value = ____________________. Small? _____
Conclude? ________________________________________
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture Practice: 10.11 p.479
L25.15
Example: Notation



Background: Want to test if mean of all male shoe
sizes could be 11.0, based on a sample mean 11.222
from 9 male students. Assume pop. s.d. 1.5.
Question: How do we denote the numbers given?
Response:




11.0 is proposed value of population mean ____
11.222 is sample mean ____
9 is sample size ____
1.5 is population standard deviation ____
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture Practice: 10.4a p.477
L25.17
Example: Intuition Before Formal Test



Background: Want to test if mean of all male shoe
sizes could be 11.0, based on a sample mean 11.222
from 9 male students. Assume pop. s.d. 1.5.
Question: What conclusion do we anticipate, by
“eye-balling” the data?
Response:
Sample mean (11.222) seems close to proposed =11.0? ___
Sample size (9) small _____________________________
S.d. (1.5) not very small ___________________________
Anticipate standardized sample mean z large? _____
P-value small? _____
conclude population mean may be 11.0? _____
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.19
Example: Test with Two-Sided Alternative



Background: Want to test if mean of all male shoe
sizes could be 11.0, based on a sample mean 11.222
from 9 male students. Assume pop. s.d. 1.5.
Question: What do we conclude from the output?
Response: z = 0.44. Large? ____
P-value (two-tailed) = 0.657. Small? ____
Conclude population mean may be 11.0? ____
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture Practice: 10.13a p.480
L25.21
P-value as Nonstandard Normal Probability
P-value is probability of sample mean as far
from 11.0 (in either direction) as 11.222.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.23
P-value as Standard Normal Probability
P-value as probability of standardized sample
mean z as far from 0 (in either direction) as 0.44.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.24
Comparing P-value Based on
vs. z
Same area under curve, just different scales on
horizontal axis due to standardizing (below).
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.25
Example: Test Results and Confidence Interval



Background: Tested if mean of all male shoe sizes could be
11.0, based on a sample mean 11.222 from 9 male students.
Assumed pop. s.d. 1.5. P-value was 0.657; didn’t reject null.
Question: Would we expect 11.0 to be contained in a
confidence interval for
?
Response: Test showed 11.0 to be plausible for  ____
(In fact, 11.0 is ______ contained in the confidence interval.)
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture Practice: 10.30b p.501
L25.27
Example: Test Results and Confidence Interval



Background: Tested if mean earnings of all
students at a university could be $5,000, based on a
sample mean $3,776 for n=446. Assumed pop. s.d.
$6,500. P-value was 0.000; rejected null hypothesis.
Question: Would 5,000 be contained in the
confidence interval for ?
Response: ____
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.29
Factors That Lead to Rejecting Ho
Statistically significant data produce P-value
small enough to reject
. z plays a role:
Reject
if P-value small; if |z| large; if…
 Sample mean far from
 Sample size n large
 Standard deviation
small
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.30
Role of Sample Size n
Large n: may reject
even if sample
mean is not far from proposed population
mean, from a practical standpoint.
Very small P-valuestrong evidence against
Ho but not necessarily very far from
.
 Small n: may fail to reject
even though
it is false.
Failing to reject false Ho is 2nd type of error.

©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.31
Definition (Review)
Type I Error: reject null hypothesis even
though it is true (false positive)
 Type II Error: fail to reject null
hypothesis even though it’s false
(false negative)
Test conclusions determine possible error:
 Reject
: correct or Type I
 Do not reject
: correct or Type II

©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.32
Example: Errors in a Medical Context
Background: A medical test is carried out for a disease
(HIV).

Questions:

What does the null hypothesis claim?

What are the implications of a Type I Error?

What are the implications of a Type II Error?

Which type of error is more worrisome?
Responses:

Null hypothesis: ________________________________

False _______: conclude _________________________

False _______: conclude _________________________

Type ___ is more worrisome.

©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture Practice: 10.64 p.513
L25.34
Example: Errors in a Legal Context
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
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Background: A defendant is on trial.
Questions:

What does
claim?

What does a Type I Error imply?

What does a Type II Error imply?

Which type is more worrisome?
Responses:

. : ________________________________

Type I: Conclude ______________________

Type II: Conclude _____________________

Type ___ is more worrisome.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.36
Sample Mean Standardizing to z
If is known, standardized follows
z (standard normal) distribution:
If
is unknown and n is large enough
(20 or 30), then
and
Can use z if is known or n is large.
What if is unknown and n is small?
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.38
Sample mean standardizing to t
For
unknown and n small,
t (like z) centered at 0 since centered at
 t (like z) symmetric and bell-shaped if
normal
 t more spread than z (s.d.>1) [s gives less info]
t has “n-1 degrees of freedom”(spread depends on n)

©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.39
Inference About Mean Based on z or t


known standardized is z
(may use z if unknown but n large)
unknown standardized is t
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.40
Inference by Hand Based on z or t
z used if
t used if
©2011 Brooks/Cole,
Cengage Learning
known or n large
unknown and n small
Elementary Statistics: Looking at the Big Picture
L25.41
z vs. t: How the Sample Mean is Standardized
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.42
Example: Distribution of t (6 df) vs. z

Background: For n=7,
has 6 df.
A Closer Look: In fact,
P(t > 2) is about 0.05;
P(z > 2) is about 0.025.


Question: How does P(t > 2) compare to P(z > 2)?
Response: P(t > 2) __________________ P(z > 2).
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.45
Practice: 10.14 p.489
Example: Distribution of t (8 df) vs. z

Background: According to 90-95-98-99 Rule for z, P(z > 2) is
between 0.01 and 0.025 because 2 is between 1.96 and 2.576.
Consider the t curve for 8 df.

Question: What is a range for P(t > 2) when t has 8 df?
Response: P(t > 2) is between ______ and ______.

©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture Practice: 10.60d p.512
L25.47
Lecture Summary
(Inference for Means: Hypothesis Tests; t Dist.)




z test about population mean: 4 steps
Examples: 1-sided and 2-sided alternatives
Relating test and confidence interval
Factors in rejecting null hypothesis




Inference based on z or t



Sample mean far from proposed population mean
Sample size large
Standard deviation small
Population sd known; standardize to z
Population sd unknown; standardize to t
Comparing z and t distributions
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L25.48